#128407

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1467
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,4,3)(5,6,9,15,11,16,12,10)(7,17,19,13,21,30,20,31)(8,22,24,28,26,14,27,25)(18,33,34,36,35,32,23,29);(5,7,10,17,12,20,16,31,11,21,15,30,9,19,6,13)(8,23,25,29,27,35,14,32,26,34,28,36,24,18,22,33);(1,3,4,2)(5,8,12,27,11,26,9,24)(6,14,10,28,16,22,15,25)(7,18,19,34,21,35,20,23)(13,29,30,33,31,36,17,32)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fgi,c^b=ch,d^b=dg,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35603);